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Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates. (English) Zbl 1244.92048

Summary: We analyze the stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R 0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist.

First we show that the disease-free equilibrium is globally asymptotically stable if and only if R 0 1. Second we show that the model is permanent if and only if R 0 >1. Moreover, using a threshold parameter R ¯ 0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1<R 0 R ¯ 0 and it loses stability as the length of the delay increases past a critical value for 1<R ¯ 0 <R 0 . Our result is an extension of the stability results of J.-J. Wang, J.-Z. Zhang and Z. Jin [Analysis of an SIR model with bilinear incidence rate. Nonlinear Anal., Real World Appl. 11, No. 4, 2390–2402 (2010; Zbl 1203.34136)].

34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models